My question is what kind of black should I use when printing. At the moment I use this black (split into CMYK):
C:0% M:0% Y:0% K:100%
It looks fine in InDesign or Illustrator but if I save a PDF file (for printing purposes) the "color" black appears more like a dark grey on screen.
My question is, if I send this file to print, the color black, on paper, will be black or something like dark grey?
It's one of the Big Things You Must Know about printing that Black (0,0,0,100) is not black; it is a dark gray. I mean really important, as in "If you don't know this, you're going to get in expensive trouble sooner or later." The reason it's gray, rather than black, is that the ink is partially absorbed by the paper and is in any case a very thin coating, so some of the white shows through.
To get actual black on press, you must create what is called a "built black" or "rich black." This is usually something like 60/60/40/100, or 40/30/30/100 as Matt mentioned. You can make a warmer black by using less cyan and more yellow, or a cooler black with more cyan and less yellow. That's a subtlety you want to get some practice with if you ever need to use it, because it can be tricky.
In InDesign and Illustrator there is a preference setting called "Appearance of Black" which can be set to display blacks "as black" or "accurately." For print work, you should always set these to display and print blacks accurately so you don't make an embarrassing error.
The main thing to remember with rich blacks is that your mix should always add up to less than 300% coverage (or whatever your printer specifies as their maximum ink coverage for the particular paper your job will be printed on). Too much ink results in "bronzing" -- a bronze sheen created by a layer of ink that can't be absorbed by the substrate, so it just sits on top. It's also almost impossible to dry completely, so you'll add time to the job.
This is why you must never use the [Registration] swatch in a layout. [Registration] is 100% everything: CMYK plus any spot color plates. Registration marks are the only place [Registration] swatch is used, but they are in the slug, not in your artwork. Your layout program does this for you automatically when you export to PDF and select the option to include registration marks.
Here's an instance where this subject can get very important: if you have a photograph with a black background and you place it on a 0/0/0/100 background, thinking they will merge invisibly, you are actually placing RGB black (which is a rich black when converted to CMYK -- check it out in Photoshop) on top of dark gray. On your screen it might look okay, but in a newspaper or magazine ad it will look pretty bad.